Action mechanism of axial flow on windage loss in open shaft‐type gap with CO2

The windage loss in rotor‐stator gap has an important effect on rotating machinery, especially with higher rotational speed and fluid density. However, the mechanism of axial flow on windage loss in open shaft‐type gap is hardly studied in most literature. To clarify it, the influences of axial Reynolds number Reu and rotational Reynolds number Reω on skin friction coefficient Cf are investigated, and flow characteristics are analyzed with different gap geometry, radius ratio η. First, the results reveal that the Cf remains constant when Reu is less than 2.8 × 104 and increases rapidly as Reu when Reu ≥ 2.8 × 104, which indicates that the effect of axial velocity u on Cf is negligible for low Reu. The positive relative deviation Δ suggests that the axial flow makes windage loss and Cf rise. Besides, a larger number of Taylor vortexes fill with gap when the effect of the centrifugal force is larger than that of the inertial force, but they gradually disappear as Reu. Subsequently, the Cf and Δ increase as η, highlighting that the effect of u on windage loss and Cf is more prominent for larger η. The fact that vorticity near walls is larger than that at the center of gap reveals that windage loss arises from the interaction between walls and fluid rather than the dissipation with fluid itself. Finally, the model of Cf in shaft‐type gap is proposed in different Reω ranges based on numerical results, and the maximum sum of squares error of 1.02 × 10−5 and minimal R2 of 0.969 satisfy the requirement of fitting accuracy and indicate that the fitting model can accurately predict Cf. The conclusions significantly help predict windage loss in open shaft‐type gap with axial flow, and further improve the design for generators of supercritical CO2 turbine‐alternator‐compressor unit.

][3][4][5] Furthermore, the density of CO 2 is relatively higher compared to the conventional working fluid, such as air or steam, as sCO 2 Brayton cycle operates at high pressure, which automatically results in exceedingly smaller sizes and higher rotational speed of rotating machinery, [6][7][8] including radial turbine, compressor, and generator.For the Brayton cycle with sCO 2 as working fluid, where the output power is below 1 MW, all rotating machines are incorporated into a sCO 2 turbinealternator-compressor (TAC) unit, [9][10][11] which can reduce the cost and enable modular construction technology. 125][16][17] It is noted that windage loss in sCO 2 TAC unit is about 37.5% of total loss, decreasing the output work of Brayton cycles. 18or the windage loss in generators or motors, Kiyota et al. 19,20 estimated and compared the windage loss in a 60 kW motor, and believed that the windage loss could make the output power decrease by 6.5%.Anderson, 21,22 through a combination of experimental and numerical studies, gave the result for windage loss, flow rate of cooling air, power, and torque of the motor versus mass flow rate in the narrow gap region of a high-speed electric motor.Liu et al. 23 explored windage loss for different structures of flux-switching permanent magnet motors, demonstrating a remarkable 70% reduction in windage loss with shrouds.
Additionally, to establish a widely applicable method for predicting windage loss and analyzing flow F I G U R E 1 Meridian plane of high-speed generator.
characteristics in generators and motors, the shaft-type gap in generators is simplified to be a concentric cylinder system, where inner walls are rotational, and outer walls are fixed.At present, numerous investigations have been conducted aimed at the windage loss in enclosed shafttype gap without axial flow.Wendlt 24 first proposed a model for predicting skin friction coefficient C f , a significant dimensionless parameter used to estimate loss in the 1930s.
Afterward, to improve the prediction accuracy and enlarge the application range of C f , Yamada, Vrancik, and Bilgen 25,26 also built the model of C f .And the result of Bilgen model is closest to numerical and experimental data compared to the other three models under ambient air conditions, based on the studies described by Deng et al. 14 and Sarri. 13owever, Hu et al. 15 further reported that the prediction accuracy of Bilgen model depends on the ranges of Re with critical CO 2 , and found that the evaluated C f by Bilgen model is more accurate when 7 × 10 3 ≤ Re ≤ 3 × 10 4 , as well as Re > 3 × 10 4 , finally modify the model of C f for Re ranging from 10 2 to 10 7 .For the windage loss in open shaft-type gap with axial flow, the related studies are extremely lacking.Currently, only the research of Zhao et al. 27 involves the effect of axial flow on windage loss, and they examined that the formation of annular vortices is inhibited by axial flow, and the influence of the rotational speed is more obvious than that of axial flow.
9][30] However, the flow in open shaft-type gap with axial flow, characterized by Taylor-Couette-Poiseuille (TCP) flow, which is formed by combining with axial flow based on TC flow, has yet to be extensively explored.Kristiawan et al. 31 experimentally investigated TCP flow under low Taylor and Reynolds numbers conditions, then discussed the influence of different flow structures on the wall shear stress components with and without axial flow, and finally found that the wall shear stress is a function of Taylor number but no dependence on Reynolds number.
According to the above descriptions, it is evident that prior investigations have concentrated on windage loss, models of C f , and TC flow in enclosed shaft-type gap, however, few researchers focused on the effects of axial flow in open shaft-type gap with CO 2 .On the other hand, the axial flow widely exists in shaft-type gap of the generators of sCO 2 TAC unit, and it can significantly make windage loss larger based on Zhao et al. 27 ; hence, it is imperative to elucidate the effects of axial flow on windage loss and TCP flow with CO 2 to predict windage loss more accurately.
Therefore, the effects of axial flow on windage loss and flow characteristics in shaft-type gap with CO 2 are investigated in detail in this paper.First, the effects of axial Reynolds number ranging from 2.8 × 10 3 to 1.4 × 10 6 and rotational Reynolds number of 10 2 -10 7 on C f are comprehensively studied.Subsequently, an in-depth analysis of the flow characteristics in shaft-type gap is conducted, with the aim of elucidating the factors influencing C f from the flow perspective.Finally, the influences of radius ratio of 0.056-0.282on C f and flow are considered.The conclusions significantly help predict windage loss in open shaft-type gap with axial flow, and further improve the design for generators of sCO 2 TAC unit.

| WINDAGE LOSS MODEL
According to Newton's law of viscosity, the shaft-type windage loss W involves several contributing factors, including the density of CO 2 ρ, the inner radius of gap R i , the angular velocity of rotor ω, as well as the length of gap L, and it is expressed as follows: where C f is the skin friction coefficient, a crucial dimensionless parameter to be exclusively investigated in Section 4. In general, under the condition of turbulence with axial flow, C f predominantly relies on rotational Reynolds number Re ω , axial Reynolds number Re u , as well as radius ratio η.In this paper, rotational Reynolds number Re ω , associated with the linear velocity of rotor, is defined as follows: where δ is the width of gap, which is equal to the differences between the outer radius of gap R o and R i , and μ is the dynamic viscosity of CO 2 .Axial Reynolds number Re u is related to the axial velocity of CO 2 and is written as: where u is the axial velocity.
Radius ratio η, a dimensionless parameter, is the ratio of δ and Ri, reflecting the geometry dimensions of the shaft-type gap: when W is studied through experiments and computational fluid dynamic (CFD) simulations, the values of torque T resulting from windage loss and ω are acquired by measurement or calculation.According to Equation (1), C f can be expressed as follows: (5) 3 | GEOMETRY AND NUMERICAL METHOD

| Geometry and boundary conditions
The schematic diagram of computational domain and the gap model consisting of stator and rotor, as presented in Figure 2, are selected as the primary objective to perform three-dimensional CFD simulations in this paper.The gap model is defined by specific parameters: the inner radius of gap R i , the length of gap L, and the inner radius of gap R o .Referring to Sarri's experimental facility, 13 which is a solid-rotor induction motor equipped with gas bearings, the values of R i and L are 35.5 and 200 mm, respectively.Besides, to make the investigations of this paper can be widely applied to the different shaft-type gap in various types of machinery, the effects of radius ratio η on skin friction coefficient C f and flow are studied; thus, the dimensions of R o need to be adjusted based on the width of gap δ, which is the variable ranging from 2.0 to 10.0 mm.Table 1 lists the values of R o , δ, and η from Geometry Ⅰ to Ⅴ.
In Figure 2, the working fluid CO 2 , whose physical properties are got from the NIST database REFPROP due to its dramatic variation near the critical point, flows through the gap from left to right.Moreover, the velocity-type boundary is specified at inlet plane, where the velocity direction of CO 2 is axial, and its magnitude ranges from 0.02 to 50 m/s to observe the influences of axial Reynolds number Re u of 2.8 × 10 3 -1.4× 10 6 on C f and flow.In addition, the inlet temperature is determined as 314.15K after considering the increase in CO2 temperature when it passes through the compressors due to various losses according to Cao et al., 32 where the inlet temperature of compressor is 305.15K, and the outlet temperature in impeller generally increases by 5-10 K for all conditions.And the outlet pressure is set as 7.38 MPa; this is because CO 2 from impeller back gap first flows through labyrinth seal cavities before leaking into shaft-type gap; thus, its pressure in the inlet of shaft-type gap ranges from 2 to 8 MPa according to the performance of labyrinth seal. 17esides, it needs to be pointed out that the pressure of 7.38 MPa at outlet plane can ensure that CO 2 in the whole shaft-type gap is supercritical state, which avoids the influence of phase transition of CO 2 on the action mechanism of axial flow on windage loss.
For every simulation, the gravity effect can be neglected as the computation domain is laid horizontally, and the outer and inner walls of gap are adiabatic and non-slip as a result of the only attention to windage loss and C f being paid.The outer wall is stationary, while the other is rotational, with its rotational speed Ω depending on rotational Reynolds number Re ω .Table 2 describes the parameters of detailed boundary.All CFD simulations are conducted using the commercial software ANSYS Fluent.The steady-state Reynoldsaveraged Navier-stokes (RANS) equations, including the continuity, momentum, and energy equations, which can provide the theory foundation for describing flow mechanics and characteristics of working fluid in gap, are solved.These equations are written as follows 33 : Continuity equation: Momentum equation: Energy equation: where Φ is defined as: In addition, in the case of turbulent flow in gap, the aforementioned equations with realizable k-ε turbulence model are solved to perform simulations in this study.Realizable k-ε turbulence model includes two unknown variables, such as turbulence kinetic energy k and the dissipation rate ε, whose transport equations are expressed as: where G k and G b are the generation terms of turbulence kinetic energy, Y M is the rate of fluctuating dilatation to overall dissipation rate, σ k , C 2 , C 1ε , C 3ε , and σ ε are constant, and C 1 is defined as follows: Furthermore, the mass flow rate m at the inlet and outlet planes is continuously monitored during the computation process.It is believed that the results are reliable when m is under the stable value condition and satisfy the criteria that the relative deviations of m at the two planes are below 1%, and the solution is converged while the residual of all the governing equations is less than 10 −6 .

| Turbulence model validation
When turbulent flow occurs, selecting an appropriate turbulence model to carefully simulate flow and accurately obtain C f in gap is essential.In this section, the three turbulence models, realizable k-ε, k-ω, and Reynolds stress model (RSM), are employed to rigorously assess the reliability of simulations.Owing to η of 0.056-0.282 in this paper containing 0.0566 for Zhao's experimental bench and the similarities in working fluid between simulations and Zhao's experiment, the experimental data obtained by Zhao et al., 27 where the dimensionless torque in shaft-type gap versus Re u is studied under axial flow conditions, is compared to CFD results using the different turbulence models, respectively.Figure 3 presents the schematic diagram of Zhao's experimental bench, where the experimental section is driven by permanent magnet motor, and the torque acting on shaft can be got by torque transducer.
The CO 2 , compressed by positive displacement pump, flows into experimental section after passing through Coriolis mass flowmeter, which facilitates the measurement of the axial velocity u of fluid.Table 3 lists the geometry dimensions and boundary conditions in Zhao et al., 27 which are in full alignment with Zhao's experiment.
The structured mesh is meticulously generated for the geometry shown in Figure 2 and specified in Table 3.However, due to the different turbulence models and near-wall treatments, which require different values of Y + , the first cell heights near walls are various.Specifically, for realizable k-ε and k-ω models, the first cell height near walls of 0.5 μm can ensure Y + closely approximates 1, whereas for the RSM, it is 5.0 μm to satisfy the requirement that Y + is larger than 30.To achieve the independence of CFD results from the mesh, the growth rate of 1.1 and the grid number of 4.32 million are utilized.
In Figure 4A,B, where the dimensionless torque T/ρν 2 L versus Re u under axial flow conditions and versus Re ω under non-axial flow conditions for each turbulence model are compared to experiment data, it can be observed that the experimental data can be evenly distributed on both sides of the numerical results, and compared to the other turbulence models, such as k-ω and RSM, the results adopting realizable k-ε model are closest to experimental data, which means that realizable k-ε model can ensure more reliable CFD results than the other models.
Furthermore, from the theory point of view, k-ω model is only applicable for the conditions of low Re ω . 34,35While realizable k-ε model is extensively used to capture flow and heat transfer details at high Re ω , 36,37 and it can be combined with enhanced wall treatment in ANSYS Fluent, making the flow in turbulent regions accurately simulated even at low Re ω conditions. 38Therefore, it is appropriate to select realizable k-ε model since Re ω ranges from 10 2 to 10 7 in this paper.Additionally, the simulations with realizable k-ε model can effectively capture velocity distribution along radial direction in shaft-type gap and predict the leakage flow rate in the sCO 2 two-teeth labyrinth seal, 39,40 further demonstrating that the realizable k-ε is reliable.| 689

| Grid independence validation
For the computational domain and gap model depicted in Figure 2, the structured mesh is generated by ANSYS ICEM, as presented in Figure 5, where the geometry dimensions are decreased to 1/4, and mesh density is coarsened to 1/8 to clearly display the grid.With meticulous attention to accurately simulating C f and flow in the boundary layer, the mesh near outer and inner walls is denser.The first cell height on walls is deliberately set as 0.5 μm, making Y + ranges from 0.025 to 0.925 under all conditions, which adherences to the prescribed Y + for the realizable k-ε model with enhanced wall treatment.Moreover, the maximum cell skewness, which falls within the prescribed range of 0.05-0.10, is used to evaluate the mesh quality.
In this section, grid independence is validated following the method recommended by Celik et al. 41 A comparative study is performed with three meshes of different grids, such as 1.78, 4.32, and 7.38 million.Figure 6A, where R denotes the radial coordinate, presents the axial velocity distribution for each mesh.Remarkably, the results are highly consistent the grid numbers increasing from 4.32 to 7.38 million, and their maximum relative deviation between both is 1.48%.
Figure 6B exhibits the solution with discretization error, and its maximum value is 3.58%.Furthermore, the analysis reveals that the maximum relative deviation between the simulated u using the 4.32 million grid and the extrapolated solution is only 1.54%, which confirms the enhanced reliability in CFD results for mesh refinement.Therefore, it is indicated that adopting 4.32 million grid can achieve a grid-independent solution based on the findings.

| RESULTS AND DISCUSSION
In this section, the influences of axial Reynolds number Re u ranging from 2.

| Effects of axial and rotational Reynolds numbers
The C f versus Re u under different Re ω of 10 2 -10 7 conditions are shown in Figure 7.It is found that the C f remains approximately constant when Re u is less than 2.8 × 10 4 , corresponding to axial Mach number Ma u , which is defined as the ratio of axial velocity u and sonic velocity, below 0.003.The reason is that u is significantly small, which is lower than 0.2 m/s when Re u is less than 2.8 × 10 4 .Under u below 0.2 m/s conditions, the u has little effect on C f and flow characteristics.However, under Re u ≥ 2.8 × 10 4 and Ma u ≥ 0.003 conditions, the C f increases rapidly as Re u .The observation suggests that the effects of u on windage loss and C f are negligible under low Re u conditions, but it becomes significantly permanent for higher Re u .The reason is that the growth of Re u and u leads to rising Ma u , thus enhancing the compressibility of working fluid and resulting in the increase of windage and C f in the gap.In addition, it is noteworthy that the C f gradually falls, and its variation rate decreases with Re ω growing from 10 2 to 10 7 , making it independent of Re ω for Re ω exceeding 10 4 , which is the same as the findings reported in Hu et al. 15 To quantitatively estimate the effects of axial flow on C f , the relative deviation Δ of C f between the existence and nonexistence of axial flow conditions (i.e., enclosed cavity) in shaft-type gap is defined as follows: where C f , n denotes skin friction coefficient under nonexistence of axial flow conditions.In Figure 8, where the relative deviation Δ are presented under Re u of 2.8 × 10 3 -1.4× 10 6 and Re ω ranging from 10 2 to 10 7 conditions, it can be seen that all values of Δ are positive whatever the values of Re u and Re ω are, which indicates that the existence of axial flow can lead to the increase of windage loss and C f rise in shaft-type gap.When Re ω = 10 2 , the values of Δ rapidly increase from 11% to 141% as Re u grows from 2.8 × 10 3 to 1.4 × 10 6 , demonstrating that higher u results in more windage loss and larger C f .The finding aligns with results and explanations provided in Figure 7.
Moreover, when Re u remains constant at 1.4 × 10 6 , the Δ remarkably decreases from 141% to 3% as Re ω rises from 10 2 to 10 7 , meaning that rotational speed of rotor Ω increases, which suggests that the influence of u on windage loss and C f is attenuated at higher Ω.The observed results can be attributed to the degree of influence of Ω on windage loss.According to the previous investigations, 14,15 the conclusion that the windage loss is approximately proportional to the cube of Ω can be drawn.Consequently, the contribution of Ω to windage loss is a minor factor, and the influence of u is equivalent to that of Ω under low Ω conditions.However, the effect of u becomes negligible in comparison to the dominant role played by Ω in generating windage loss at high Ω.In fact, the conclusion in Figure 8 may be further validated in Kim et al., 42 where Eq. ( 24) describes that C f is proportional to the 0.0091 power of Re u under the Ω of rotor of 35 000 r/min conditions, corresponding to Re ω = 2.4 × 10 6 , which indicates the influences of axial flow on windage loss and larger C f is minor at higher Re ω and Ω.
The mass flow rate m plotted against Re ω under Re u of 2.8 × 10 3 -1.4× 10 6 conditions are described in Figure 9. Remarkably, it is noted that the m remains constant for Re ω < 10 6 , suggesting that the m is unaffected by Ω; however, it slightly declines when Re ω is larger 10 5 , as the influence of centrifugal force on the flow becomes pronounced under high Ω conditions, making the vortexes generated in shaft-type gap inhibit m.Besides, both m and its variation rate grow with Re u ; hence the influence of Re u on m is more prominent under the conditions of higher Re u , which is attributed to the higher u of working fluid.contours at meridional plane with Re ω of 10 2 and 10 , as well as Re u ranging from 2.8 × 10 3 to 1.4 × 10 6 are presented in Figure 10.

| Flow characteristic analysis
In the streamline diagrams presented, it is found that there is no vortex in shaft-type gap while Re ω is 10 2 , and CO 2 can flow smoothly throughout the whole computational domain, regardless of the values of Re u .This is because the fluid is minimally affected by the centrifugal force arising from the rotation of rotor, but the inertial force resulting from the axial flow continuously plays the dominant role at low Re ω .
However, for Re ω = 10 7 , it can be seen that the gap becomes filled with a larger number of Taylor vortexes due to centrifugal force, which is related to flow state and geometry dimension, and their characteristics have been described in Hu et al., 15,16  | 693 increase of Re u , corresponding to rising u, the number of Taylor vortexes decreases, and the flow is characterized by the of axial flow and Taylor vortexes, as evidenced streamlines under Re u = 1.4 × 10 5 conditions.Moreover, a notable transition occurs, where the Taylor vortexes entirely disappear, while Re u ≥ 2.8 × 10 5 , signifying the complete dominance of the inertial force in influencing the flow characteristics.
In the velocity contour, a notable trend, where the average velocity rises with Re u and Re ω increasing, is observed, which results from concurrent increases in u and Ω. Besides, when Re ω = 10 2 , the velocity at the center of gap is larger, but as Re ω research 10 7 , it gradually decreases from rotational wall to stationary wall.According to the fact that the velocity near rotational wall equals the linear velocity of rotor, and it is 0 near fixed wall due to the imposition of nonslip boundary conditions.Consequently, the u is the primary factor influencing velocity distribution at low Re ω , and the Ω has more significant influence on velocity at high Re ω .To provide a more comprehensive understanding of the velocity distribution near walls, Figure 11 depicts the velocity at different radial positions under Re ω = 10 2 and Re u of 2.8 × 10 3 -1.4× 10 6 conditions.
In Figure 11, the velocity exhibits a remarkable increase near the rotational wall, contrasting with a decrease near the stationary wall.It needs to be pointed out that the gradient of velocity near rotational wall is far smaller compared to that near stationary wall, which is attributed to the thicker boundary layer near rotational wall, arising from the higher Re ω .Additionally, due to axial flow, the velocity in this paper is different from the previous investigations in Hu et al., 15 where the maximum and minimum velocity occurs near rotational and stationary walls, respectively.
In Figure 12, where temperature contours at inlet and out planes with Re ω of 10 2 and 10 7 , as well as Re u of 1.4 × 10 6 are shown, it can be seen that the temperature at outlet is higher than that at inlet when Re ω is 10 2 or 10 7 since the generated windage loss due to viscous effect can heat CO 2 passing through shaft-type gap.In addition, the temperature for Re ω = 10 7 is also higher than that for Re ω = 10 2 ; this is because the windage loss is larger for higher Re ω , corresponding to higher Ω, which results in more heat transferred to the fluid.The conclusions are reasonable and consistent with the law of conservation of energy.Besides, it is noted that temperature variation is smaller from inlet to outlet compared to other conventional working fluid, such as air or steam, due to the larger specific heat capacity of high-pressure CO 2 , which is beneficial to the safe and long-time operation of sCO 2 TAC unit.
In Figure 13, where the pressure contours in the whole gap with Re ω of 10 2 and 10 7 , as well as Re u of 1.4 × 10 6 are described, it can be found that the pressure decrease from inlet to outlet due to the existence of axial flow, and it is larger than the critical pressure of 7.38 MPa in whole gap since the outlet pressure for all conditions is determined as 7.38 MPa.Therefore, it is believed that there is no phase transition of CO 2 in gap, which leads to little influence on flow characteristics and windage loss.
The streamlines and velocity contours with Re ω of 10 7 and Re u of 2.8 × 10 3 in the three-dimension geometry are shown in Figure 14.It is found that the whole gap is filled with a considerable number of vortexes, which can be clearly presented in the meridional plane, under the conditions of high Re ω and low Re u because the centrifugal force due to the rotation of rotor is far larger than inertia force, which is also consistent with the F I G U R E 11 Velocity distribution at different radial positions under Re ω = 10 2 and Re u of 2.8 × 10 3 -1.4× 10 6 conditions.conclusion in Figure 10.In addition, it is noted that the velocity of CO 2 is constant from the three-dimension streamlines, but there is the small number of regions of velocity due to the existence of Taylor vortex.

| Effects of radius ratio
To analyze the effects of different η, corresponding to Geometry Ⅰ-V listed in Table 1, on windage loss, the C f versus η under the condition of Re u ranging from 2.8 × 10 3 to 1.4 × 10 6 at Re ω = 10 2 and Re ω = 10 7 are presented in Figure 15, respectively.As shown in Figure 15A,B, it is evident that the C f linearly increases as η whether Re ω is 10 2 or 10 7 , suggesting that the windage loss and C f grow as the width of gap δ, which is as a result of increasing m passing through gap for larger δ.
According to the comparison to Figure 7, the influence of Re u and Re ω on windage loss and C f is more significant than that of η.Therefore, the flow characteristics, dependent on Ω and u, emerge as the decisive factor for windage loss rather than geometry dimension, such as η.Furthermore, the C f increases as Re u , while it decreases with the increase in Re ω , which is the same as the conclusion in Figure 7, and the reasons will further be analyzed in Figure 17.conditions at Re ω = 10 2 and Re ω = 10 7 , respectively.In Figure 16, it is noted that the Δ grows with η, for example, under Re ω = 10 2 and u = 1.4 × 10 6 conditions, the values of Δ increase from 141% at η = 0.056 to 200% at η = 0.282.This finding further highlights that the effect of u on windage loss and C f is more prominent under larger η conditions.Based on the analysis, the fact that when the η rises, namely δ is larger, the influencing ranges of Ω tend to be smaller, but that of u is more dominant in affecting windage loss and C f , is the primary reason.
Furthermore, it is worth mentioning that when Re ω = 10 7 , all values of Δ are below 20%, which is far lower than that at Re ω = 10 2 , further demonstrating that the effect of u on C f can decrease as Ω climbs, which is consistent with the conclusion in Figure 8.
To analyze the influences of flow characteristics with the η range of 0.056-0.282,the vorticity contours from Geometry Ⅰ to Ⅴ at meridional plane with Re u of 2.8 × 10 3 , 1.4 × 10 5 , and 1.4 × 10 6 at Re ω = 10 2 are presented in Figure 17.
For all contours, there are large vorticity regions near walls, and the direction of vorticity near rotational and stationary walls is opposite.Meanwhile, the vorticity at the center of gap is close to 0, revealing that windage loss arises predominantly from the interaction between the wall and the fluid rather than dissipation with the fluid itself.Additionally, according to previous studies, 16 it is found that C f is influenced by rough walls because the grain was in blending or logarithmic law regions occurred, which impacted the flow in boundary layer, which also indicates that the interaction of walls on windage loss and C f can not be ignored.Of course, the conclusion also reveals that lowering surface roughness of all walls is one approach to reducing windage loss in the gap for engineering applications.
Moreover, the large vorticity region increases as Re u , making the dissipation effect in gap enhances, and subsequently, larger windage loss and C f , aligning with Figure 7 and Figure 15.Conversely, the observation that large vorticity region decreases as δ, which results from the larger δ makes the number of vortex generated in gap reduce due to its growing diameter.Nevertheless, despite the decrease in vorticity, the windage loss and C f still increase with η due to the increase in m through gap for larger δ.To numerically predict C f in shaft-type gap under axial flow conditions, the model of C f is proposed based on the numerical results in this paper, can provide an accurate prediction model of windage loss during the design and operation of sCO 2 TAC units.Firstly, according to the analysis from Sections 4.1 to 4.3, it is well known that C f is related to Re ω , η, and Re u ; thus, the model of C f can be expressed as follows: ).
ω u f (14)   Additionally, it needs to be pointed out that the model of C f can be related to Re u and Taylor number Ta, which is defined as follows, and is the criterion that Taylor vortices due to the centrifugal force affecting the fluid particles occur.
It is noted that Ta is related to Reω and η, which indicates that C f is still dependent on Re u , Re ω , and η when its model is the functions of Re u and Ta in Ref. 42 And in this paper, the influences of Re u , Re ω , and η on C f are investigated from Sections 4.1 to 4.3; Thus, it is reasonable and appropriate to adopt Equation ( 14) to propose the model of C f .Subsequently, from Figures 8 and 16, it is found that the influences of axial flow on C f vary when Re ω is in different ranges; hence, the determination of the model of C f as a piecewise function with Re ω of 10 3 and 10 5 as the demarcation point is acceptable to improve the prediction accuracy of the model.
Finally, referring to Bilgen model, 26 C f is expressed as an exponential function, which is written as: where a is the correction coefficient of the model of C f , b, c, and d are the index of Re ω , η, and Re u , which are constant and can be determined by numerical results in this paper.
Adopting the least-square method, the model of C f is proposed, as listed in Table 4.It is worth noting that the developed model is only applicable to the range of Re ω of 10 2 -10 6 , η of 0.056-0.282,and Re u of 2.8 × 10 3 -1.4× 10 6 .
Furthermore, Sum of Squares Error SSE and R-square R 2 , which are used to evaluate the accuracy of fitting model, are defined as: where S i is the regression value, and s i is the sample value.
where s ¯i s i is the average sample value.
From the above equations, it is apparent that SSE closer to 0 and R 2 approaching 1 indicate the model of C f is more accurate.For all equations in different ranges of Re ω in Table 4, the maximum SSE and minimal R 2 are 1.02 × 10 −5 and 0.969, which can satisfy the requirement of fitting accuracy and indicate that the appropriate model of C f is close to the numerical results and accurately predicts C f in shaft-type gap with axial flow.

| CONCLUSIONS
The windage loss in rotor-stator gap has an important effect on rotating machinery, especially with higher rotational speed and fluid density.However, the mechanism of axial flow on windage loss in open shaft-type gap is hardly studied in most literature.To clarify it, the influences of axial Reynolds number Re u and rotational Reynolds number Re ω on skin friction coefficient C f are investigated, and flow characteristics are analyzed with different gap geometry, radius ratio η.Finally, the model of C f in shafttype gap is proposed in different Re ω ranges based on numerical results.The conclusions help predict windage loss in open shaft-type gap with axial flow, and further improve the design for generators of supercritical CO 2 TAC unit.The main conclusions are drawn as follows: 1.The C f remains constant when Re u is less than 2.8 × 10 4 and increases rapidly as Re u when Re u ≥ 2.8 × 10 4 , which indicates that the effect of axial velocity u on C f is negligible under low Re u conditions.The positive relative deviation Δ suggests that the axial flow makes the windage loss and C f rise, and the Δ rapidly increases as Re u grows, demonstrating that higher u results in and larger C f , and it remarkably decreases from 141% to 3% with ω rising, which means the influence of u on C f is attenuated at higher rotational speed Ω. 2. At Re ω of 10 2 , there is no vortex in shaft-type gap regardless of the values of Re u since the inertial force resulting from the axial flow plays the dominant role at low Re ω .However, for Re ω = 10 7 , a larger number of Taylor vortexes fill with the whole shaft-type gap when Re u ≤ 2.8 × 10 4 , due to larger effect of centrifugal force than that of inertial force, but their size gradually decreases under Re u = 1.4 × 10 5 conditions and disappears while Re u ≥ 2.8 × 10 5 .Besides, the velocity remarkably increases near the rotational wall, contrasting with a decrease near the stationary wall.3. The C f and Δ increase as η whether Re ω is 10 2 or 10 7 , such as the values of Δ increasing from 141% at η = 0.056 to 200% at η = 0.282, which highlights that the effect of u on windage loss and C f is more prominent under larger η conditions.Furthermore, the fact that vorticity near walls is far larger than that at the center of gap, which is close to 0, reveals that windage loss arises from the interaction between walls and fluid rather than the dissipation with the fluid itself.

F I G U R E 2
Diagram of computational domain and gap model composed of stator and rotor.

T A B L E 3
Geometry dimensions and boundary conditions in Zhao et al.27

F
I G U R E 4 Comparison between numerical and Zhao's experimental results: (A) is dimensionless torque versus Re u , and (B) is dimensionless torque versus Re ω .F I G U R E 5 Mesh of gap model (geometry dimensions decreased to 1/4 and mesh density coarsened to 1/8).
8 × 10 3 to 1.4 × 10 6 and rotational Reynolds number Re ω of 10 2 -10 7 on skin friction coefficients C f are comprehensively studied.Subsequently, an analysis of the flow characteristics in shaft-type gap is conducted, with the aim of elucidating the factors influencing C f from the flow perspective.Subsequently, the effects of radius ratio η of 0.056-0.282on C f and flow are focused on.Finally, the model of C f in shaft-type gap is proposed in different Re ω ranges based on numerical results.

F
I G U R E 6 Grid-independence validation: (A) is axial velocity distribution, and (B) is solution with discretization error.F I G U R E 7 Skin friction coefficient C f for different Re u under the conditions of Re ω of 10 2 -10 7 .
To comprehensively investigate the influences of Re ω and Re u on flow characteristics, the streamlines and velocity F I G U R E 8 Relative deviation under Re u of 2.8 × 10 3 -1.4× 10 6 and Re ω of 10 2 -10 7 conditions.F I G U R E 9 Mass flow rate plotted against Re ω under Re u of 2.8 × 10 3 -1.4× 10 6 conditions.
at Re u ≤ 2.8 × 10 4 , indicating that effects of the centrifugal force acting on working fluid is far larger than that of the inertial force.With the F I G U R E 10 Streamlines and velocity contours with Re ω of 10 2 and 10 7 , as well as Re u of 2.8 × 10 3 -1.4× 10 6 .HU ET AL.

Figure 16
represents the values of Δ under η of 0.056 ~0.282 and Re u ranging from 2.8 × 10 3 to 1.4 × 10 6F I G U R E 12 Temperature contours at inlet and outlet planes with Re ω of 10 2 and 10 7 , as well as Re u of 1.4 × 10 6 .F I G U R E 13 Pressure in the whole gap with Re ω of 10 2 and Re u of 1.4 × 10 6 .F I G U R E 14 Streamlines and velocity contours with Re ω of 10 7 and Re u of 2.8 × 10 3 .

T A B L E 4 10 5 <
Model of C f .Re ω ≤ 10 7

4 .
The model of C f , which can provide an accurate prediction model of windage loss during the design and operation of sCO 2 TAC units, is proposed based on the numerical results in this paper.To improve prediction accuracy of the model, the model is a piecewise function with Re ω of 10 3 and 10 5 as the demarcation point.Moreover, the maximum SSE of 1.02 × 10 −5 and minimal R 2 of 0.969 satisfy the requirement of fitting accuracy and indicate that the fitting model can accurately predict C f in shaft-type gap with axial flow.
Values of the outer radius of gap R o , width of gap δ, and radius ratio η.Boundary conditions of the computational domain.
T A B L E 1